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International Journal of Mathematics and Mathematical Sciences
Volume 22, Issue 2, Pages 391-400
http://dx.doi.org/10.1155/S0161171299223915

Remarks on μ-measurbale sets: regularity, σ-smootheness, and measurability

Pace University, Pace Plaza, New York 10038, NY, USA

Received 10 September 1996; Revised 3 March 1998

Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let X be an arbitrary nonempty set and a lattice of subsets of X such that ϕ,X. 𝒜() is the algebra generated by and () denotes those nonnegative, finite, finitely additive measures μ on 𝒜(). I() denotes the subset of () of nontrivial zero-one valued measures. Associated with μI() (or Iσ()) are the outer measures μ and μ considered in detail. In addition, measurability conditions and regularity conditions are investigated and specific characteristics are given for 𝒮μ, the set of μ-measurable sets. Notions of strongly σ-smooth and vaguely regular measures are also discussed. Relationships between regularity, σ-smoothness and measurability are investigated for zero-one valued measures and certain results are extended to the case of a pair of lattices 1,2 where 12.